# Algebra of Complex Numbers

**Addition**

Properties of addition

(1) Sum of two complex numbers is a complex number (closure law)

(3)

(4) 0 + 0*i*denoted by 0 is the identity for addition (0 is the additive identity) since*z*+ 0 = 0 +*z*=*z*. For any complex number*z*.

(5) -*z*= -*a*-*ib*is the negative or additive inverse of*z*=*a*+*ib*

*. Since**z,*-*z*is the additive inverse*z.*- Subtraction:
is defined as

- Multiplication of complex numbers

Properties of multiplication

(1) Product of two complex numbers is a complex number (closure law)

(2)

(3)

(4) 1 + 0*i*is the multiplicative identity since

(5) multiplicative inverse or reciprocal of*z*. Since we cannot leave a complex number in the denominator of a rational number, we can simplify by multiplying and dividing by its conjugate.

Note:- Division of a complex number by another is done using the same principle.

**Example 7:**

**Solution**:-

**Example**: If ,prove that

**Solution**:

Using property (1) of conjugates

**Example 9**: If , find the real values of

*x*and

*y*.

**Solution**:

Equating real parts on both sides,

Equating imaginary parts on both sides

Squaring (1)

Substitute for

*y*from (2)

Solution set

**Example 10**

**:**

P represents a variable complex number Z. Find the locus of P if

**Solution**

**:**

Let [treat it like point (

*x*,

*y*)]

(Remark: you will learn soon that a complex number can bne represented by a point in a plane.)

Given

This is the locus of

**Identities as applied to complex numbers**

In fact all the identities which are true for real numbers are true for complex numbers also.